optimisation problem in algebra

Question

Max $x_1x_2x_3x_4$ sub to $ x_1+x_2+x_3+x_4=5 , x_i>=0$ One way i can think of is to reduce the max function in 3 variables using the constraint and then the regular maxima minima procedures. But the hint in the question says " use that $(x-k)(x+k)<=k^2$. Does the problem relate to AM GM inequality? i am unable to relate the question and the hint. Any help would be very good.

Answer 1

It indeed can be taken care of with the A.M.-G.M. inequality: if you already know (or can prove) that

Let $a_1,\dots,a_n $ be positive numbers, then

(i) $\enspace\: G(a_1,\dots, a_n)=\bigl(a_1\dotsm a_n\bigr)^{\tfrac1n}\le A(a_1,\dots, a_n)= \dfrac{a_1+\dots+a_n}n$;

(ii)$\enspace$ This inequality is an equality if and only if $a_1=\dots =a_n$.

This says the maximum is attained for $x_1=x_2=x_3=x_4=4$, and it is $$\frac{x_1+x_2+x_3+x_4}4=\frac 54.$$ Therefore the maximum of the product is $$x_1+x_2+x_3+x_4=\biggl(\frac 54\biggr)^{\mkern-5mu4}=\frac{625}{256}.$$